Analytic Geometry is a branch of mathematics that treats the relation of algebraic functions and their respective graphs, or pictures that can be drawn from these functions. Students are first introduced to analytic geometry in Algebra II courses, and delve further into its study in both pre-calculus and calculus courses. Essentially, this branch of mathematics combines geometry and algebra to show what certain mathematical relationships, called functions, look like in the real world.



There are four well known shapes--also known as curves or functions in math jargon-- that students spend a good time studying in analytic geometry. These are the circle, parabola, ellipse, and hyperbola. These curves are called conic sections because they can be derived in a most curious way from a figure called a double-napped cone. A double-napped cone is formed by placing two cones on top of each other, balancing at the tip. Essentially what you end up with is something that looks like an hourglass made out of two ice cream cones--without the ice cream, of course, or the sand. By appropriately slicing the double-napped cone with a plane, we end up with each of the different conic sections.

Of the four conic sections, indubitably the circle is the one with which most people are familiar. One does not have to look to far to find applications of the circle in everyday life, although many of these are taken for granted. The circle, by mathematical definition, is the set of all points which are an equal distance from another fixed point. If you got that the fixed distance is the radius and the fixed point is the center of the circle, take a bow. Well done!



Now what makes the circle so special and why? Well, it is the precise definition of this conic section that has caused this shape to have such a monumental effect on the progress of mankind. For how would man have progressed without the wheel? Precisely because the points of the circle are all the same distance from the center, is the smooth roll of the wheel possible. After all, how efficient would your car or dune buggy be, for that matter, if the wheels were in the shape of hexagons? Ah, I bet you never thought of that. Indeed it is hard to imagine that hungry car salesman making a good pitch for that new car model with those strange hexagonal tires!



Another point that comes to mind regarding circles is why manhole covers are round. Think about this in context of what you now know about circles, and it becomes apparent why this unique shape works for this application. Why would a cover in the shape of a pentagon (five-sided figure) not work here? If the answer is not immediate, think of what might happen to the poor utility worker standing at the bottom of the hole, while his careless coworker fiddled with the cover above.



Conic sections. Yes. Now we see how the precise mathematical definition of a shape as omnipresent as the circle plays a critical role in determining the useful applications of this curiously round shape. Next time you see a circle, think of how lucky you are that your tires are round and not hexagonal. Stay tuned...

About the author:

Joe is a prolific writer of self-help and educational material. Under the penname, JC Page, Joe authored the classic of mathematical ABC's Arithmetic Magic. Joe is also author of the charmingly pithy and popular ebook, Make a Good Impression Every Time: The Secret to Instant Popularity; the seminal collection of verse, Poems for the Mathematically Insecure. For more information, visit his website at www.mathbyjoe.com.